Properties

Label 315.2.j.e
Level 315315
Weight 22
Character orbit 315.j
Analytic conductor 2.5152.515
Analytic rank 00
Dimension 44
Inner twists 22

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Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [315,2,Mod(46,315)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(315, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 2, names="a")
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("315.46"); S:= CuspForms(chi, 2); N := Newforms(S);
 
Level: N N == 315=3257 315 = 3^{2} \cdot 5 \cdot 7
Weight: k k == 2 2
Character orbit: [χ][\chi] == 315.j (of order 33, degree 22, minimal)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [4,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(2)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: no
Analytic conductor: 2.515287663672.51528766367
Analytic rank: 00
Dimension: 44
Relative dimension: 22 over Q(ζ3)\Q(\zeta_{3})
Coefficient field: Q(2,3)\Q(\sqrt{2}, \sqrt{-3})
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: x4+2x2+4 x^{4} + 2x^{2} + 4 Copy content Toggle raw display
Coefficient ring: Z[a1,,a4]\Z[a_1, \ldots, a_{4}]
Coefficient ring index: 1 1
Twist minimal: no (minimal twist has level 35)
Sato-Tate group: SU(2)[C3]\mathrm{SU}(2)[C_{3}]

qq-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the qq-expansion are expressed in terms of a basis 1,β1,β2,β31,\beta_1,\beta_2,\beta_3 for the coefficient ring described below. We also show the integral qq-expansion of the trace form.

f(q)f(q) == q+(β2+β1+1)q2+(2β3+β2+2β1)q4+(β2+1)q5+(β3β2+β1)q7+(β33)q8+(β3+β2+β1)q10++(3β3+9β2+7β1+8)q98+O(q100) q + (\beta_{2} + \beta_1 + 1) q^{2} + (2 \beta_{3} + \beta_{2} + 2 \beta_1) q^{4} + (\beta_{2} + 1) q^{5} + ( - \beta_{3} - \beta_{2} + \beta_1) q^{7} + (\beta_{3} - 3) q^{8} + (\beta_{3} + \beta_{2} + \beta_1) q^{10}+ \cdots + (3 \beta_{3} + 9 \beta_{2} + 7 \beta_1 + 8) q^{98}+O(q^{100}) Copy content Toggle raw display
Tr(f)(q)\operatorname{Tr}(f)(q) == 4q+2q22q4+2q5+2q712q82q10+4q118q13+4q146q16+4q174q208q222q232q2512q2622q28+4q29+12q31++14q98+O(q100) 4 q + 2 q^{2} - 2 q^{4} + 2 q^{5} + 2 q^{7} - 12 q^{8} - 2 q^{10} + 4 q^{11} - 8 q^{13} + 4 q^{14} - 6 q^{16} + 4 q^{17} - 4 q^{20} - 8 q^{22} - 2 q^{23} - 2 q^{25} - 12 q^{26} - 22 q^{28} + 4 q^{29} + 12 q^{31}+ \cdots + 14 q^{98}+O(q^{100}) Copy content Toggle raw display

Basis of coefficient ring in terms of a root ν\nu of x4+2x2+4 x^{4} + 2x^{2} + 4 : Copy content Toggle raw display

β1\beta_{1}== ν \nu Copy content Toggle raw display
β2\beta_{2}== (ν2)/2 ( \nu^{2} ) / 2 Copy content Toggle raw display
β3\beta_{3}== (ν3)/2 ( \nu^{3} ) / 2 Copy content Toggle raw display
ν\nu== β1 \beta_1 Copy content Toggle raw display
ν2\nu^{2}== 2β2 2\beta_{2} Copy content Toggle raw display
ν3\nu^{3}== 2β3 2\beta_{3} Copy content Toggle raw display

Character values

We give the values of χ\chi on generators for (Z/315Z)×\left(\mathbb{Z}/315\mathbb{Z}\right)^\times.

nn 127127 136136 281281
χ(n)\chi(n) 11 1β2-1 - \beta_{2} 11

Embeddings

For each embedding ιm\iota_m of the coefficient field, the values ιm(an)\iota_m(a_n) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   ιm(ν)\iota_m(\nu) a2 a_{2} a3 a_{3} a4 a_{4} a5 a_{5} a6 a_{6} a7 a_{7} a8 a_{8} a9 a_{9} a10 a_{10}
46.1
−0.707107 1.22474i
0.707107 + 1.22474i
−0.707107 + 1.22474i
0.707107 1.22474i
−0.207107 0.358719i 0 0.914214 1.58346i 0.500000 + 0.866025i 0 −1.62132 2.09077i −1.58579 0 0.207107 0.358719i
46.2 1.20711 + 2.09077i 0 −1.91421 + 3.31552i 0.500000 + 0.866025i 0 2.62132 + 0.358719i −4.41421 0 −1.20711 + 2.09077i
226.1 −0.207107 + 0.358719i 0 0.914214 + 1.58346i 0.500000 0.866025i 0 −1.62132 + 2.09077i −1.58579 0 0.207107 + 0.358719i
226.2 1.20711 2.09077i 0 −1.91421 3.31552i 0.500000 0.866025i 0 2.62132 0.358719i −4.41421 0 −1.20711 2.09077i
nn: e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
7.c even 3 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 315.2.j.e 4
3.b odd 2 1 35.2.e.a 4
7.c even 3 1 inner 315.2.j.e 4
7.c even 3 1 2205.2.a.n 2
7.d odd 6 1 2205.2.a.q 2
12.b even 2 1 560.2.q.k 4
15.d odd 2 1 175.2.e.c 4
15.e even 4 2 175.2.k.a 8
21.c even 2 1 245.2.e.e 4
21.g even 6 1 245.2.a.g 2
21.g even 6 1 245.2.e.e 4
21.h odd 6 1 35.2.e.a 4
21.h odd 6 1 245.2.a.h 2
84.j odd 6 1 3920.2.a.bv 2
84.n even 6 1 560.2.q.k 4
84.n even 6 1 3920.2.a.bq 2
105.o odd 6 1 175.2.e.c 4
105.o odd 6 1 1225.2.a.k 2
105.p even 6 1 1225.2.a.m 2
105.w odd 12 2 1225.2.b.h 4
105.x even 12 2 175.2.k.a 8
105.x even 12 2 1225.2.b.g 4
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
35.2.e.a 4 3.b odd 2 1
35.2.e.a 4 21.h odd 6 1
175.2.e.c 4 15.d odd 2 1
175.2.e.c 4 105.o odd 6 1
175.2.k.a 8 15.e even 4 2
175.2.k.a 8 105.x even 12 2
245.2.a.g 2 21.g even 6 1
245.2.a.h 2 21.h odd 6 1
245.2.e.e 4 21.c even 2 1
245.2.e.e 4 21.g even 6 1
315.2.j.e 4 1.a even 1 1 trivial
315.2.j.e 4 7.c even 3 1 inner
560.2.q.k 4 12.b even 2 1
560.2.q.k 4 84.n even 6 1
1225.2.a.k 2 105.o odd 6 1
1225.2.a.m 2 105.p even 6 1
1225.2.b.g 4 105.x even 12 2
1225.2.b.h 4 105.w odd 12 2
2205.2.a.n 2 7.c even 3 1
2205.2.a.q 2 7.d odd 6 1
3920.2.a.bq 2 84.n even 6 1
3920.2.a.bv 2 84.j odd 6 1

Hecke kernels

This newform subspace can be constructed as the kernel of the linear operator T242T23+5T22+2T2+1 T_{2}^{4} - 2T_{2}^{3} + 5T_{2}^{2} + 2T_{2} + 1 acting on S2new(315,[χ])S_{2}^{\mathrm{new}}(315, [\chi]). Copy content Toggle raw display

Hecke characteristic polynomials

pp Fp(T)F_p(T)
22 T42T3++1 T^{4} - 2 T^{3} + \cdots + 1 Copy content Toggle raw display
33 T4 T^{4} Copy content Toggle raw display
55 (T2T+1)2 (T^{2} - T + 1)^{2} Copy content Toggle raw display
77 T42T3++49 T^{4} - 2 T^{3} + \cdots + 49 Copy content Toggle raw display
1111 T44T3++16 T^{4} - 4 T^{3} + \cdots + 16 Copy content Toggle raw display
1313 (T2+4T4)2 (T^{2} + 4 T - 4)^{2} Copy content Toggle raw display
1717 T44T3++16 T^{4} - 4 T^{3} + \cdots + 16 Copy content Toggle raw display
1919 T4+8T2+64 T^{4} + 8T^{2} + 64 Copy content Toggle raw display
2323 T4+2T3++1 T^{4} + 2 T^{3} + \cdots + 1 Copy content Toggle raw display
2929 (T1)4 (T - 1)^{4} Copy content Toggle raw display
3131 (T26T+36)2 (T^{2} - 6 T + 36)^{2} Copy content Toggle raw display
3737 T4 T^{4} Copy content Toggle raw display
4141 (T210T+17)2 (T^{2} - 10 T + 17)^{2} Copy content Toggle raw display
4343 (T210T+23)2 (T^{2} - 10 T + 23)^{2} Copy content Toggle raw display
4747 (T22T+4)2 (T^{2} - 2 T + 4)^{2} Copy content Toggle raw display
5353 T4+8T3++64 T^{4} + 8 T^{3} + \cdots + 64 Copy content Toggle raw display
5959 T4+8T3++3136 T^{4} + 8 T^{3} + \cdots + 3136 Copy content Toggle raw display
6161 T46T3++3969 T^{4} - 6 T^{3} + \cdots + 3969 Copy content Toggle raw display
6767 T4+22T3++14161 T^{4} + 22 T^{3} + \cdots + 14161 Copy content Toggle raw display
7171 (T28T56)2 (T^{2} - 8 T - 56)^{2} Copy content Toggle raw display
7373 T4+4T3++16 T^{4} + 4 T^{3} + \cdots + 16 Copy content Toggle raw display
7979 T4+24T3++18496 T^{4} + 24 T^{3} + \cdots + 18496 Copy content Toggle raw display
8383 (T2+2T161)2 (T^{2} + 2 T - 161)^{2} Copy content Toggle raw display
8989 T4+6T3++529 T^{4} + 6 T^{3} + \cdots + 529 Copy content Toggle raw display
9797 (T212T+4)2 (T^{2} - 12 T + 4)^{2} Copy content Toggle raw display
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